Simplify the following expression and state the condition under which the simplification is valid. You can assume that $y \neq 0$. $q = \dfrac{7y}{4(y - 7)} \div \dfrac{y}{7(y - 7)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{7y}{4(y - 7)} \times \dfrac{7(y - 7)}{y} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 7y \times 7(y - 7) } { 4(y - 7) \times y } $ $ q = \dfrac{49y(y - 7)}{4y(y - 7)} $ We can cancel the $y - 7$ so long as $y - 7 \neq 0$ Therefore $y \neq 7$ $q = \dfrac{49y \cancel{(y - 7})}{4y \cancel{(y - 7)}} = \dfrac{49y}{4y} = \dfrac{49}{4} $